"... In this paper we give two equivalent characterizations of the Caucal hierarchy, a hierarchy of infinite graphs with a decidable monadic second-order (MSO) theory. It is obtained by iterating the graph transformations of unfolding and inverse rational mapping. The first characterization sticks to thi ..."

In this paper we give two equivalent characterizations of the Caucal hierarchy, a hierarchy of infinite graphs with a decidable monadic second-order (MSO) theory. It is obtained by iterating the graph transformations of unfolding and inverse rational mapping. The first characterization sticks to this hierarchical approach, replacing the language-theoretic operation of a rational mapping by an MSO-transduction and the unfolding by the treegraph operation. The second characterization is non-iterative. We show that the family of graphs of the Caucal hierarchy coincides with the family of graphs obtained as the ε-closure of configuration graphs of higher-order pushdown automata. While the different characterizations of the graph family show their robustness and thus also their importance, the characterization in terms of higher-order pushdown automata additionally yields that the graph hierarchy is indeed strict.

"... We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by model-theoretic interpretations. ..."

We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by model-theoretic interpretations.

"... This is a survey on the relationship between logical definability and computational complexity on finite structures. Particular emphasis is given to game-based evaluation algorithms for various logical formalisms and to logics capturing complexity classes. In addition to the ..."

This is a survey on the relationship between logical definability and computational complexity on finite structures. Particular emphasis is given to game-based evaluation algorithms for various logical formalisms and to logics capturing complexity classes. In addition to the

"... Abstract. It is a well-known result that the set of reachable stack contents in a pushdown automaton is a regular set of words. We consider the more general case of higher-order pushdown automata and investigate, with a particular stress on effectiveness and complexity, the natural notion of regular ..."

Abstract. It is a well-known result that the set of reachable stack contents in a pushdown automaton is a regular set of words. We consider the more general case of higher-order pushdown automata and investigate, with a particular stress on effectiveness and complexity, the natural notion of regularity for higher-order stacks: a set of level k stacks is regular if it is obtained by a regular sequence of level k operations. We prove that any regular set of level k stacks admits a normalized representation and we use it to show that the regular sets of a given level form an effective Boolean algebra. In fact, this notion of regularity coincides with the notion of monadic second order definability over the canonical structure associated to level k stacks. Finally, we consider the link between regular sets of stacks and families of infinite graphs defined by higher-order pushdown systems.

"... We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic second-order (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of eleme ..."

We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic second-order (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of elements of the orignal structure. The definition of these interpretations directly implies that they send structures with a decidable WMSO theory to structures with a decidable first-order theory. In this paper, we investigate the expressive power of such interpretations applied to infinite deterministic trees. The results can be used in the study of automatic and tree-automatic structures.

by
Dietrich Kuske, Markus Lohrey
- TO APPEAR IN ANNALS OF PURE AND APPLIED LOGIC

"... We prove that a finitely generated group is context-free whenever its Cayleygraph has a decidable monadic second-order theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of context-free groups and also proves a conjecture of Schupp. To derive this re ..."

We prove that a finitely generated group is context-free whenever its Cayleygraph has a decidable monadic second-order theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of context-free groups and also proves a conjecture of Schupp. To derive this result, we investigate general graphs and show that a graph of bounded degree with a high degree of symmetry is context-free whenever its monadic second-order theory is decidable. Further, it is shown that the word problem of a finitely generated group is decidable if and only if the first-order theory of its Cayley-graph is decidable.

Abstract. Model-checking problems for PDL (propositional dynamic logic) and its extension PDL ∩ (which includes the intersection operator on programs) over various classes of infinite state systems (BPP, BPA, pushdown systems, prefix-recognizable systems) are studied. Precise upper and lower bounds are shown for the data/expression/combined complexity of these model-checking problems. 1

"... We define the class of micro-macro stack graphs, a new class of graphs modeling infinite-state sequential systems with a decidable model-checking problem. Micro-macro stack graphs are the configuration graphs of stack automata whose states are partitioned into micro and macro states. Nodes of the gr ..."

We define the class of micro-macro stack graphs, a new class of graphs modeling infinite-state sequential systems with a decidable model-checking problem. Micro-macro stack graphs are the configuration graphs of stack automata whose states are partitioned into micro and macro states. Nodes of the graph are configurations of the stack automaton where the state is a macro state. Edges of the graph correspond to the sequence of micro steps that the automaton makes between macro states. We prove that this class strictly contains the class of prefix-recognizable graphs. We give a direct automata-theoretic algorithm for model checking ¢-calculus formulas over micro-macro stack graphs. 1

"... We prove that a connected graph of bounded degree with only finitely many orbits has a decidable MSO-theory if and only if it is context-free. This implies that a group is context-free if and only if its Cayley-graph has a decidable MSO-theory. On the other hand, the rst-order theory of the Cayl ..."

We prove that a connected graph of bounded degree with only finitely many orbits has a decidable MSO-theory if and only if it is context-free. This implies that a group is context-free if and only if its Cayley-graph has a decidable MSO-theory. On the other hand, the rst-order theory of the Cayley-graph of a group is decidable if and only if the group has a decidable word problem. For Cayley-graphs of monoids we prove the following closure properties. The class of monoids whose Cayley-graphs have decidable MSO-theories is closed under free products. The class of monoids whose Cayley-graphs have decidable first-order theories is closed under general graph products. For the latter result on first-order theories we introduce a new unfolding construction, the factorized unfolding, that generalizes the tree-like structures considered by Walukiewicz. We show and use that it preserves the decidability of the first-order theory. Most of

"... infinite structures. Abstract. The theorem of factorisation forests shows the existence of nested factorisations — a la Ramsey — for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the importance of this approach in th ..."

infinite structures. Abstract. The theorem of factorisation forests shows the existence of nested factorisations — a la Ramsey — for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the importance of this approach in the context of automata over infinite words and trees. We extend the theorem of factorisation forest in two directions: we show that it is still valid for any word indexed by a linear ordering; and we show that it admits a deterministic variant for words indexed by well-orderings. A byproduct of this work is also an improvement on the known bounds for the original result. We apply the first variant for giving a simplified proof of the closure under complementation of rational sets of words indexed by countable scattered linear orderings. We apply the second variant in the analysis of monadic second-order logic over trees, yielding new results on monadic interpretations over trees. Consequences of it are new caracterisations of prefix-recognizable structures and of the Caucal hierarchy. 1